Set minimum distance between hair particles
After researching a bit further, it looks like this is not a built-in feature for Blender. Although there are some workarounds to sort of achieve the effect that have been mentioned in the comments, the solution seems to be an add-on:

https://blenderartists.org/forum/showthread.php?233643-Addon-Moleculars-physics

http://pyroevil.com/molecular-script-download/

Here is a quick GIF of me doing particle collision with the add-on (I am on a Mac, so I am using an older version):



This question is very similar:

How to correct collision between particles